Polynomials and Completing the Square
In this video, we’re going to look at polynomials and discuss some ways to solve polynomial equations. Now, the first thing we need to do is define what a polynomial is. A polynomial is an expression in one variable where the variable is raised to a higher power than 1, or it has terms that are raised to a higher power than 1.
One example of a polynomial might be x to the fourth plus x to the third plus x squared plus x equals 1. This would be our polynomial here. This whole thing is a polynomial equation, because we’re saying that this equation, or this expression here, is equal to some constant.
We have—it’s a polynomial, because we have terms in x that are higher powers than just 1. We have x squared, x cubed, x to the fourth power. Most of the polynomials that we’ll be looking at and concentrating on here will be polynomials that just have powers of x squared and lower. Even these ones that are higher than x squared are still polynomials.
We just won’t be looking at them as closely. Primarily, we’ll be looking at polynomials that look kind of like this. We have an x squared term, an x term, and a constant. If you’ll notice just based on what we have here, we have x squared plus x equals 1. It’s not immediately obvious how we would solve this.
We can’t use all the same tricks that we did when we just had x to the power of 1. We have an x squared and an x. We can’t really combine them and we can’t just take a square root, because if we just tried to take a square root, we’d end up taking the square root of x squared plus x, which is significantly more complicated than we already have.
We’re stuck with some special methods that we have to use to solve polynomial equations. Let’s look at the example: x squared plus 4x equals 12. We’re going to do what’s called “completing the square” here. Completing the square is something that we can do that will allow us to take a square root cleanly.
To do that, what we need to do is we need to add some constant, we’ll call it c, to both sides, such that we can rewrite the equation in the form “x plus b squared equals a”. What we need to do is we need to pick our c to add to this side of the equation, such that we can write that side as x plus b squared.
Now, when you have x squared plus some coefficient times x, b is going to be this coefficient divided by two. In this case, what we’re going to want to have here is x plus 2 squared. In order to get x plus 2 squared on this side, we have to figure out what c will be based on x plus 2 squared.
If we have x plus 2 squared, that’s the same as x plus 2 times x plus 2. We’ll go ahead and multiply this out, so we see that we get. x times x is x squared. x times 2 is 2x. 2 times x is 2x, and 2 times 2 is 4. This gives us x squared plus 4x plus 4. What this tells us is that our c that we need to add to both sides of the equation is 4.
Coming back up to here, we have x squared plus 4x plus 4 equals 12 plus 4. We’ve seen from our calculations up here that x squared plus 4x plus 4 is the same as x plus 2 squared. We can rewrite this as x plus 2 squared equals 16. Now that we’ve got just a single term on both sides here, we can take the square root of both sides.
We’ll take the square root here, and that just gives us x plus 2, since x plus 2 squared—the square root of x plus 2 squared is just x plus 2. On this side, we have the square root of 16, which is 4, but because we took the square root of both sides, we don’t know if the square root that we’re talking about here is plus 4 or negative 4.
We have to write plus or minus 4, because it could be either one in the context of what we’re doing here. x plus 2 is equal to plus or minus 4, or x is equal to -2 plus or minus 4. We have two solutions to this polynomial up here. It’s either minus 2 plus 4, or minus 2 minus 4. There are two solutions to this polynomial equation: x equals 2 and x equals -6.
When you have a polynomial equation like this, generally you’re going to get two solutions. They’ll usually be separated by whatever this term here is, because you’ll have your base number and then you’ll have the possibility if you add this number and the possibility if you subtract this number.
Occasionally, what you’ll find is that this number is equal to 0. Then, there is only one solution. In that case, you will just have one solution. This is the completing the square method for solving polynomial equations.